MA530_JAN09 - = { x + iy : | x | < / 4 ,-...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 530 Qualifying exam January 2009 (A. Eremenko) Each problem is worth 10 points. You can use any theorem proved in class or a theorem from the textbook if you state it completely and correctly. 1. Let f n be a sequence of injective analytic functions in the open unit disc, and suppose that f = lim n →∞ f n uniformly on compact subsets of the unit disc. Prove that f is either injective or constant. 2. Let Ω be a bounded region in the plane and f : Ω Ω an analytic function that maps Ω into itself. Suppose that there exists a point z 0 Ω such that f ( z 0 ) = z 0 . Prove that | f 0 ( z 0 ) | ≤ 1. Hint: If a function maps something into itself, iterating it is a good idea . 3. Consider the Taylor expansion in a neighborhood of the point i : z cot z = X n =0 c n ( z - i ) n . What is the radius of convergence of the series in the right hand side? 4. Let f be an analytic function in the strip
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = { x + iy : | x | < / 4 ,- < y < } , and suppose that | f ( z ) | 1 and f (0) = 0. Prove that | f ( z ) | | tan z | for all z . 5. Find a harmonic function in the region { z : | z | < 1 , Im z > } whose boundary values are 1 on the interval (-1 , 1) and 0 on the half-circle. 6. Let f and g be two analytic functions in some region D , and suppose that f ( z ) + g ( z ) is real for all z D . Prove that f-g is constant. 7. For all real a , evaluate Z tan( x + ia ) dx. 1 Use the principal value when a = 0. 8. Let f be a meromorphic function in a neighborhood of 0 and an analytic function in a neighborhood of 0 with the properties (0) = 0 and (0) 6 = 0. Prove that res f = res [( f ) ] . 2...
View Full Document

This document was uploaded on 01/25/2012.

Page1 / 2

MA530_JAN09 - = { x + iy : | x | < / 4 ,-...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online